The PostScript graphics model itself does not support general transparency of page elements at all. (Hence it is also not possible for EPS.) PostScript colors are all fully opaque.

An object drawn on top of another object would overwrite and cover all lower objects with its own color leaving no room for transparent effects. (If you see something that looks like transparency overlays in a PostScript viewer or printout, then that was only emulated transparency, by flattening the two (or more) respective objects into one single rasterized area creating the illusion of transparency.)

The PDF graphics model is based on PostScript's, but it extends it in various aspects, adding several new features. One of these is real transparency for complete objects.

After Adobe added transparency to PDF, it also created an extension [1] to the existing PostScript language that was able to include code in PS programs which would add transparency to PDFs created from this PostScript via Distiller. However, when rendering on screen or printing on paper this same original PostScript including this same code, that additional transparency would not appear, and the top (transparent in PDF) object would still overwrite the bottom ones when directly used in PostScript.

What other choice would I have?

Various:

Use PDF only. Don't use EPS.

If you must use EPS, use a two-step process:

Create the PDF first.

Then convert from the (transparency-enabled) PDF to EPS, 'flattening' the transparent elements into rasterized areas which emulate the desired transparency effect.

[1] The name of this extension is called pdfmark. With the help of the pdfmark operator one can also add other features to PostScript code which only materialize when distilling this PostScript to PDF: annotations, interactive form fields and buttons, metadata, hyperlinks, and more. All these elements would not have any effect in the direct PostScript rendering on screen or on paper prints.